Advertisement Remove all ads

If a and b are two vectors such that |a+b|=|a|, then prove that vector 2a+b is perpendicular to vector b - Mathematics

Advertisement Remove all ads
Advertisement Remove all ads
Advertisement Remove all ads

If `veca and vecb` are two vectors such that `|veca+vecb|=|veca|,` then prove that vector `2veca+vecb` is perpendicular to vector `vecb`

 

Advertisement Remove all ads

Solution

Given |`veca+vecb|=|veca|`∣

`|veca+vecb|^2=|veca|^2`

`|veca|^2+2veca.vecb+|vecb|^2=|veca|^2`

`2veca.vecb+|vecb|^2=0 ................(1)`

`Now (2veca+vecb)(vecb)=2vecavecb+vecbvecb=2vecavecb+|vecb|^2=0  using(1)`

We know that, if the dot product of two vectors is zero, then either of the two vectors is zero or the vectors are perpendicular to each other.

Thus,

Given `|veca+vecb|=|veca|`

`|veca+vecb|^2=|veca|^2`

`|veca|^2+2veca.vecb+|vecb|^2=|veca|^2`

`2veca.vecb+|vecb|^2=0 ................(1)`

`Now (2veca+vecb)(vecb)=2vecavecb+vecbvecb=2vecavecb+|vecb|^2=0  using(1)`

We know that, if the dot product of two vectors is zero, then either of the two vectors is zero or the vectors are perpendicular to each other.

Thus, `2veca+vecb` is perpendicular to `vecb`

 

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
  Is there an error in this question or solution?
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×